An Explicit Time-Difference Scheme with an Adams–Bashforth Predictor and a Trapezoidal Corrector

2012 ◽  
Vol 140 (1) ◽  
pp. 307-322 ◽  
Author(s):  
Sajal K. Kar
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Models ◽  
Time Difference ◽  
Von Neumann ◽  
Fully Implicit ◽  
Implicit Schemes ◽  
Spatial Dimensions ◽  
Von Neumann Stability ◽  
Predictor Corrector

Abstract A new predictor-corrector time-difference scheme that employs a second-order Adams–Bashforth scheme for the predictor and a trapezoidal scheme for the corrector is introduced. The von Neumann stability properties of the proposed Adams–Bashforth trapezoidal scheme are determined for the oscillation and friction equations. Effectiveness of the scheme is demonstrated through a number of time integrations using finite-difference numerical models of varying complexities in one and two spatial dimensions. The proposed scheme has useful implications for the fully implicit schemes currently employed in some semi-Lagrangian models of the atmosphere.

scholarly journals Efficiency of High-Order Accurate Difference Schemes for the Korteweg-de Vries Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Kanyuta Poochinapan ◽  
Ben Wongsaijai ◽  
Thongchai Disyadej
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Kdv Equation ◽  
Numerical Models ◽  
Finite Difference Methods ◽  
Von Neumann ◽  
The Kdv Equation ◽  
Von Neumann Analysis ◽  
Numerical Tools ◽  
Finite Difference Techniques

Two numerical models to obtain the solution of the KdV equation are proposed. Numerical tools, compact fourth-order and standard fourth-order finite difference techniques, are applied to the KdV equation. The fundamental conservative properties of the equation are preserved by the finite difference methods. Linear stability analysis of two methods is presented by the Von Neumann analysis. The new methods give second- and fourth-order accuracy in time and space, respectively. The numerical experiments show that the proposed methods improve the accuracy of the solution significantly.

scholarly journals New schemes for a two-dimensional inverse problem with temperature overspecification

2001 ◽  
Vol 7 (3) ◽  
pp. 283-297 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Central Processor ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fully Implicit ◽  
Implicit Schemes ◽  
Alternating Direction

Two different finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the (3,3) alternating direction implicit (ADI) finite difference scheme and the (3,9) alternating direction implicit formula. These schemes are unconditionally stable. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett [17]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit schemes for two-dimensional diffusion with temperature overspecification. The alternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their unconditional stability is significant. The results of numerical experiments are presented, and accuracy and the Central Processor (CPU) times needed for each of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.

scholarly journals An Accurate Predictor-Corrector-Type Nonstandard Finite Difference Scheme for an SEIR Epidemic Model

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Asma Farooqi ◽  
Riaz Ahmad ◽  
Rashada Farooqi ◽  
Sayer O. Alharbi ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Continuous System ◽  
Euler Method ◽  
State Variables ◽  
Nonstandard Finite Difference Scheme ◽  
Long Time ◽  
Predictor Corrector ◽  
Nonstandard Finite Difference

The present work deals with the construction, development, and analysis of a viable normalized predictor-corrector-type nonstandard finite difference scheme for the SEIR model concerning the transmission dynamics of measles. The proposed numerical scheme double refines the solution and gives realistic results even for large step sizes, thus making it economical when integrating over long time periods. Moreover, it is dynamically consistent with a continuous system and unconditionally convergent and preserves the positive behavior of the state variables involved in the system. Simulations are performed to guarantee the results, and its effectiveness is compared with well-known numerical methods such as Runge–Kutta (RK) and Euler method of a predictor-corrector type.

scholarly journals A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

2020 ◽  
Vol 52 (3) ◽  
pp. 322-338
Author(s):  
Nasrin Okhovati ◽  
Mohammad Izadi
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Discontinuous Coefficients ◽  
Flow In Porous Media ◽  
Two Phase ◽  
Flux Function ◽  
Uniform Grids ◽  
Discontinuous Flux ◽  
Predictor Corrector

In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented.

A predictor–corrector compact finite difference scheme for a nonlinear partial integro-differential equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shufang Hu ◽  
Wenlin Qiu ◽  
Hongbin Chen
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Integro Differential Equation ◽  
Compact Finite Difference Method ◽  
Backward Euler ◽  
Predictor Corrector

Abstract A predictor–corrector compact finite difference scheme is proposed for a nonlinear partial integro-differential equation. In our method, the time direction is approximated by backward Euler scheme and the Riemann–Liouville (R–L) fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, a two-step predictor–corrector (P–C) algorithm called MacCormack method is used. A fully discrete scheme is constructed with space discretization by compact finite difference method. Numerical experiment presents the scheme is in good agreement with the theoretical analysis.

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